Question: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 8x + 21$, and $ m \angle MON = 3x + 3$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {8x + 21} + {3x + 3} = {90}$ Combine like terms: $ 11x + 24 = 90$ Subtract $24$ from both sides: $ 11x = 66$ Divide both sides by $11$ to find $x$ $ x = 6$ Substitute $6$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 8({6}) + 21$ Simplify: $ {m\angle LOM = 48 + 21}$ So ${m\angle LOM = 69}$.